Convex bodies with equipotential circles

Abstract

Given a convex body $K\subset \mathbb R^2$ we say that a circle $\Omega\subset \text{int} \ K$ is an equipotential circle if every tangent line of $\Omega$ cuts a chord $AB$ in $K$ such that for the contact point $P=\Omega\cap AB$ it holds that $|AP|\cdot|PB|=\lambda$, for a suitable constant number $\lambda$. The main result in this article is the following: Let $K\subset\mathbb R^2$ be a convex body which has an equipotential circle $\mathcal B$ with centre $O$ in its interior. Then $K$ has centre of symmetry at $O$, moreover, if none chord of $K$ which is tangent to $\mathcal B$ subtends an angle $\pi/2$ from $O$, then $K$ is a disc. We also derive some results which characterizes the ellipsoid and the sphere in $\mathbb R^3$ and introduce also the concept of equireciprocal disc.